The ABC triangle is inscribed in a circle centered at the point O.

The ABC triangle is inscribed in a circle centered at the point O. Find the degree measure of the angle C of the ABC triangle if the BAO angle is 59 °.

Given:
Inscribed triangle ABC.
Point O, center of the circle.
BAO angle = 59 °
To find:
BCA angle -?
Decision:
Draw a straight line from point O to point B.
AO = OB = r can be seen from the figure.
It follows that the triangle is isosceles. Means:
Angle OAB = Angle OBA = 59
Angle AOB = 180 – (OAB + OBA) = 180 – (59 + 59) = 180 – 118 = 62
Angle AOB is the center angle that rests on the arc AB.
Arc AB = 62
And the angle ACB is the inscribed angle and is equal to half of the arc on which it rests.
Angle ACB = 62: 2 = 32